Contemporary Mathematics

Moduli Spaces of Homotopy Theory

David Blanc

This article is dedicated to the memory of Bob Brooks, colleague and friend

Abstract. The moduli spaces refered to are topological spaces whose pathcomponents parametrize homotopy types. Such objects have been studied intwo separate contexts: rational homotopy types, in the work of several authorsin the late 1970’s; and general homotopy types, in the work of Dwyer-Kan andtheir collaborators. We here explain the two approaches, and show how theymay be related to each other.

1. Introduction

The concept of “moduli” for a mathematical object goes back to Riemann,who used it to describe a set of parameters determining the isomorphism class ofa Riemann surface of a given topological type. He also recognized that the setof all conformal equivalence classes of such surfaces can itself be given a complexstructure, although this was only made precise by Teichmuller, Ahlfors, Bers, andothers in the twentieth century. Similarly, the family of birational equivalenceclasses of algebraic curves of each genus (over C, say) are parametrized by analgebraic variety of moduli, as Mumford showed, and there are analogues in higherdimensions.

If we fix an oriented surface S of genus g, a marked Riemann surface is a choiceof an orientation-perserving diffeomorphism of S into a compact Riemann surfaceX , and the equivalence classes of such marked surfaces form Teichmuller space Tg,a complex analytic variety. The mapping class group Γg of orientation preservingdiffeomorphisms of S acts on Tg, and the quotient is isomorphic to the modulispace Mg of isomorphism classes of complex structures on S – or equivalently,of smooth projective curves over C. Since the action of Γg is virtually free, andTg is contractible, Mg has the same rational homology as the classifying spaceBΓg.

Note that the set of all Riemann surfaces of a given genus can be obtained bydeforming a given conformal (or equivalently, complex) structure on a fixed sample

1991 Mathematics Subject Classification. Primary 55P15; Secondary 55Q99, 55P62, 18G30,14J10.

c©2004 David Blanc

37

38 DAVID BLANC

surface Σg – or the corresponding Fuchsian group Γ. Such deformations aretypically governed by an appropriate cohomology group.

Note also that there are several levels of possible structures on a compactsurface: topological, differentiable, real or complex analytic, hyperbolic, conformal,metric, and so on, which define different types of moduli space. See [31] or [51] formore details and references on the classical theory.

1.1. Moduli spaces. Thus, there are a number of common themes in themoduli problems arising in various areas of mathematics:

(a) Finding a set of parameters to describe appropriate equivalence classes ofobjects in a certain geometric category. These may occur at varous levels,and we distinguish between discrete parameters (such as the genus of asurface) and the finer (continuous) moduli.

(b) Giving the set of such equivalence classes the structure of an object in thesame category, which we may call a moduli space.

(c) The deformation of a given structure, controlled by cohomology, as ameans of obtaining all other possible structures.

(d) To pass between levels of structure, one can often quotient by a groupaction.

As we shall see, all these themes will find expression in the context of homotopytheory. One aspect of the usual moduli spaces for which we have no analogue inour setting is the important role played by compactifications.

1.2. Moduli spaces in homotopy theory. The homotopy theory of theclassical moduli spaces of Riemann surfaces and the corresponding mapping classgroups has been studied extensively – see, for example, [29, 35, 53]. However,this is not our subject; we shall be concerned here rather with the analogues ofsuch moduli spaces in the category of topological spaces, answering to the generaldescription given above.

The simplest examples of this approach are provided by various mapping spaces,starting with the loop space ΩX of all pointed maps from the circle into a topolog-ical space X . Here the set π0ΩX of components is isomorphic to the fundamentalgroup of X , and the higher homotopy groups are again those of X (re-indexed).More significantly, the components of space of maps from X to BO or BU,the classifying spaces of the infinite orthogonal or unitary groups, correspond toequivalence classes of (real or complex) vector bundles over X , and the higher ho-motopy groups correspond to various reductions of structure, by Bott periodicity.Further examples of simple “moduli spaces” of topological spaces are provided bythe configuration spaces of n distinct points in a manifold M , which have also beenstudied extensively (see [21] for a comprehensive survey).

1.3. Moduli spaces and homotopy types. Our focus here will be morespecifically on moduli spaces of all homotopy types (suitably interpreted), whichhave been investigated from at least two points of view over the past twenty-fiveyears:

• In work of Lemaire-Sigrist, Felix and Halperin-Stasheff, simply-connectedrational homotopy types correspond to the components of a certain (infi-nite dimensional) algebraic variety over Q – or alternatively, the quotientof such a variety by appropriate group actions (Sec. 2).

MODULI SPACES OF HOMOTOPY THEORY 39

• The “classification complex” approach of Dwyer-Kan yields a space whosecomponents parametrize the homotopy types of all CW complexes (Sec. 6).

Of course, neither statement has much content, as stated; the point is that bothapproaches offer (surprisingly similar) tools for inductively analyzing the collectionof all relevant homotopy types:

In each case we begin with a coarse classification of homotopy types by algebraicinvariants, such as the cohomology algebra H of a space, or the correspondingstructure on homotopy groups, called a Π-algebra (which reduces rationally toa graded Lie algebra over Q – see Sec. 3). One might also refine this initialclassification using both homotopy and cohomology groups (Sec. 5).

(a) For each H , we have homological classification of spaces with this co-homology algebra. This is usually presented as an obstruction theory,stated in terms of (algebraically defined) cohomology groups of H . It canbe reinterpreted as expressing the moduli space as the limit of a towerof fibrations (Sec. 7), with successive fibers described in terms of thesecohomology groups (Sec. 8).

(b) In the rational case, we have a deformation theory approach, which de-scribes all homotopy types with H∗(X, Q) ∼= H by perturbing a canonicalmodel. In fact, this was the main focus of the rational homotopy theoryapproach mentioned above.

Our main goal here is to bring out the connection between the integral andrational cases. From the first point of view, this can be made explicit at the alge-braic level, comparing the two cohomology theories through appropriate spectralsequences (Sec. 9).

On the other hand, the rational deformation theory has no integral analogue,since it uses differential graded models, which exist only for simply-connected ratio-nal homotopy types. However, it turns out that the deformations can be describedgeometrically in terms of higher homotopy operations (Sec. 4), and these do haveintegral versions.

1.4. Remark. One could – with some justice – argue that the analogybetween the rational and integral “spaces of homotopy types” we describe hereand classical moduli spaces is rather tenuous (although it is merely incidental tothe point we want to make). One aspect in particular for which we can offer noreal analogue is the fact that the classical moduli spaces have the same structure(algebraic or analytic variety, etc.) as the objects being classified.

In some sense, however, a homotopy type is just the collection of all its homo-topy invariants, suitably interpreted (see [41, I,4.9]). In fact, one goal of homotopytheory is to produce a manageable set of such invariants, sufficient for distinguish-ing between type. A starting point for such a set of invariants is the Postnikovsystem of a space (and its invariants); so it is perhaps fitting that our analysis ofthe space of all homotopy types is carried out by means of its Postnikov system, andthe associated collections of homotopy invariants (higher homotopy operations, orcohomology classes) which appear in our obstruction theory. Thus one could chooseto view these collections of invariants themselves – rather than the space of ho-motopy types – as the true “moduli object”. This is certainly the best way tounderstand the connections between the rational and integral cases, and it also

40 DAVID BLANC

might make (somewhat far-fetched) sense of the claim that the moduli object is it-self of the same kind as the things it classifies. We leave the philosophically-inclinedreader to pursue this line of thought at his or her discretion.

1.5. Acknowledgements. I would like to thank the referee for his or hercomments, and Bill Dwyer, Paul Goerss, Jim Stasheff, and Yves Felix for helpfulconversations and remarks on various aspects of the subject.

2. Rational homotopy and deformation theory

We first describe the deformation-theoretic approach to rational homotopytypes, starting with some basic background material:

2.1. Rational homotopy. Rational homotopy theory deals with rational al-gebraic invariants of homotopy types – that is, it disregards all torsion in thehomology and homotopy groups. Quillen and Sullivan, respectively, proposed twomain algebraic models for the rational homotopy type of a simply-connected spaceX :

(i) A differential graded-commutative algebra, or DGA (A∗X , d), where

(AiX)∞i=0 is a graded-commutative algebra over Q, equipped with a differ-

ential d of degree +1 (which is a graded derivation with respect to theproduct), and H∗(AX , d) ∼= H∗(X, Q). Since X is simply-connected, wemay assume (A∗

X , d), is, too – that is, A0X = Q, A1

X = 0.

(ii) A differential graded Lie algebra, or DGL (LX , ∂), with LX = (LXi )∞i=1

a positively-graded Lie algebra over Q, and ∂ a graded Lie derivation ofdegree −1, such that Hi(L

X , ∂) ∼= πiΩX ⊗ Q for i ≥ 1, with theSamelson products as Lie brackets.

For more information see the survey [25], and the original sources [42, 50].

2.2. Definition. A DGA of the form (ΛV, d), where ΛV is the free graded-commutative algebra generated by the graded vector space V = (V i)∞i=0 is calledcofibrant. A cofibrant DGA (ΛV, d) such that Im (d) ⊆ Λ+V ·Λ+V (where Λ+Vis the sub-algebra of elements in positive degree), is called a minimal model.

A DGA or DGL map which induces an isomorphism in (co)homology is calleda quasi-isomorphism.

2.3. Proposition (see [1]). Any simply-connected DGA A has a minimal modelwith a quasi-isomorphism q : (ΛV, d) → (A, dA), and this is unique up to isomor-phism.

One can also define cofibrant and minimal models for DGLs, and show thatthey have similar properties (cf. [25, §21]).

2.4. Remark. Both the category DGA1 of simply-connected DGAs, and thecategory DGL0 of connected DGLs have model category structures (see [41]);thus each has a concept of homotopy, and as Quillen showed, the correspondinghomotopy categories hoDGA1 and hoDGL0 are both equivalent to the rationalhomotopy category hoT Q

1 of simply-connected topological spaces (cf. [42]). The

weak equivalences in T Q1 correspond of course to quasi-isomorphisms for DGAs

or DGLs.

MODULI SPACES OF HOMOTOPY THEORY 41

Note that every simply-connected graded-commutative algebra over Q is real-izable as the rational cohomology ring of some (1-connected) space, and similarly,every positively graded Lie algebra is realizable as the rational homotopy groups ofa space.

2.5. The space of rational homotopy types. We may therefore identify

the collection M = MT Q1 of all rational weak homotopy types with the set of weak

homotopy types of 1-connected DGAs (over Q) – i.e., equivalence classes underthe equivalence relation generated by the quasi-isomorphisms – and similarly forDGLs. Moreover, there is a coarse classification of such types, with the discreteparametrization provided by the cohomology rings; we denote the sub-collection ofM consisting of all DGAs with a given graded cohomology ring H by MH . Thishas a distinguished element:

2.6. Definition. A DGA A is called formal if it is quasi-isomorphic to (H∗A, 0)(the cohomology ring of A thought of as a DGA with zero differential). Dually, aDGL L is coformal if it is quasi-isomorphic to (H∗L, 0).

MH was studied by Jean-Michel Lemaire and Francois Sigrist in [34], by YvesFelix in [23], and independently by Steve Halperin and Jim Stasheff in [27]. Theyshowed that every rational homotopy type in MH can be obtained by suitabledeformations of a fixed model of the unique formal object:

2.7. Definition. The bigraded model (B, d) for H is the minimal modelB = ΛZ for (H, 0) of Proposition 2.3, where Z = Z∗

∗ , the graded vector spaceof generators, is equipped with an additional homological grading (indicated by thelower index).

The differential takes Zin to Λ(⊕n−1

k=1Zk)i+1, so one still has a homologicalgrading on the cohomology groups of B. The vector spaces Z∗

n are defined (almostcanonically) by induction on n, so that at the n-th stage we kill the cohomology inhomological dimension k for 0 < k < n. (ΛZ∗, d) is essentially the Tate-Jozefiakresolution of a graded commutative algebra H (cf. [52, 32]) – i.e., a minimal“cellular” resolution. See [27, §3] for the details.

The deformation consists of a perturbation of the differential d to D = d + D′

– of course, this is done in such a way that the cohomology is unchanged, andD D = 0. We can no longer expect D to respect both gradings, but if we think ofB as being only filtered by FnB := ⊕n

k−0 (ΛZ∗)k, then the perturbation is suchthat D′ takes Zn into Fn−2B.

2.8. Theorem (cf. [27, Theorem 4.4]). If (A, dA) is any DGA with H∗(A, dA)∼= H, one can deform the bigraded model (B, d) for (H, 0) into a filtered model(B, D) quasi-isomorphic to (A, dA), and this is unique up to isomorphism.

2.9. The variety of deformations. The deformations D of d are determinedby successive choices of linear transformations D′

n : Zn → Fn−2ΛZ∗<n, satisfying

a quadratic algebraic condition (to ensure that D2 = 0). Thus the collection of allpossible perturbations of the given bigraded model (B, d) constitute an (infinitedimensional) algebraic variety over Q, which we denote by VH . Every simply-connected rational homotopy type is represented by at least one point of VH ; eachsuch point actually represents an “augmented” DGA X , equipped with a specificisomorphism φ : H∗(X ; Q) → H .

42 DAVID BLANC

However, one can have more than one point corresponding to each augmentedrational homotopy type: the points of MH correspond bijectively with the orbitsof VH under the action of an appropriate group, as in [34, Thm. 4] or [23, Ch. 1](note that these two superificially different descriptions are in fact the same, by[24]):

In the approach described in [47], we first divide by the action of a certainpro-unipotent algebraic group GH to obtain the set VH/GH of augmentedrational homotopy types as above. We must then further mod out by Aut(H)to get the plain (i.e., unaugmented) rational homotopy types, so that MH

∼=Aut(H)\VH/GH . In [47, §6] it is then shown that the quotient set VH/GH canalso be described as the set of path components of the Cartan-Chevalley-Eilenberg“standard construction” C(L) (cf. §5.2) on a certain Lie algebra L (namely, thealgebra of derivations of the bigraded model for H).

Note that though MH can be uncountable, it can also be small: a DGA(A, d) is called intrinsically formal if MH∗A consists of a single point.

2.10. Obstruction theory. We have seen that the collection MT Q1 of all

simply-connected rational homotopy types indeed exhibits the basic characteristicsof a moduli space listed in §1.1. However, to make this useful, we need furtherinformation about its overall structure.

This is provided by Halperin and Stasheff in the form of an obstruction theoryfor realizing a given isomorphism of rational cohomology algebras φ : H∗(A, dA) →H∗(B, dB) by a DGA map (or equivalently, by a map of the corresponding rationalspaces). Since such a map induces a homotopy equivalence between the respectivecofibrant models, this can be thought of as a method for distinguishing betweenthe homotopy types in MH .

The obstruction theory, which only works in full generality if the DGAs inquestion are of finite type, was presented in [27, §5] in terms of a sequence ofelements On(f) ∈ Hom(Zn+1, H(B)) – or rather, in a quotient of this group. In[49, §4], these are re-interpreted as elements in

⊕p Hp+1(B; πp(H(A))).

2.11. Remark. By analogy with classical deformation theory, we would expectthe deformations of the bigraded model to be “controlled” by a suitable differentialgraded Lie algebra (L, d), or perhaps by its cohomology. In fact, Felix observedthat the Halperin-Stasheff obstructions On(f) can be thought of as lying in theF-cohomology group F H1

n(B, d) of the bigraded model (B, d) for (H, 0) (see[23, Ch. 5, Prop. 7]). Moreover, these groups can be identified with the tangent(or Harrison) cohomology of the graded algebra H (cf. [23, Ch. 3, Prop. 4]), whichis the same as the Andre-Quillen cohomology, and this can indeed be calculated asthe cohomology of a suitable DGL (cf. [46, §4]).

Moreover, both Schlessinger and Stasheff (see [49, §5]) and Felix (cf. [23, Ch. 4-5]) obtain results connecting formality and intrinsic formality of a DGA with itstangent cohomology, though the relation does not work both ways (see [23, Ch. 4,Ex. 2] ).

We shall give a uniform treatment of this cohomological description in a moregeneral context in Section 8. However, for technical reasons (inter alia, convergenceof second- versus first-quadrant spectral sequences), this is done for the Eckmann-Hilton dual – that is, in terms of Π-algebras (the integral version of graded Lie

MODULI SPACES OF HOMOTOPY THEORY 43

algebras over Q). The mod p version – that is, an obstruction theory for realizingunstable (co)algebras over the Steenod algebra – appears in [6].

3. Moduli spaces of graded Lie algebras

The description of the moduli space of rational homotopy types in terms ofDGLs, rather than DGAs, has received less attention. However, as noted above, ithas certain technical advantages, especially in the integral case (where we need nolonger restrict to simply-connected spaces, incidentally). The dualization to DGLsis quite straightforward, but we shall describe it here from a slightly different pointof view.

3.1. Deformations of DGLs. The deformation theory for DGLs proceedsprecisely along the lines indicated in §2.5ff.: given a positively-graded Lie algebraP = P∗ over Q, consider the collection MP of all rational homotopy typesof simply-connected spaces X – or equivalently, of DGLs (L, ∂) – such thatπ∗ΩX ⊗ Q = H∗(L, ∂) ∼= P (as Lie algebras).

Again there is a distinguished coformal element (P, 0) in MP (Def. 2.6). ItsDGL minimal model (B, ∂) can be given an additional homological grading, andthe differential in this bigraded model can be deformed so as to obtain a filteredmodel for any X in MP . See the sketch in [28]; the details appear in theunpublished thesis [40].

3.2. The resolution model category. Since the category DGL0 is notabelian, from the point of view of homotopical algebra it is more natural to con-sider simplicial resolutions of objects in this category, rather than “chain complex”resolutions (which is what the homological grading of the bigraded model gives us).Moreover, this generalizes more naturally to the integral setting, where differentialgraded models are no longer available (although see [36] for a way around this).

Recall that a simplicial object X• over a category C is a functor X : ∆op → C,where ∆ is the category of finite sequences n = (0, 1, . . . , n) (n ∈ N), with order-preserving maps as morphisms. The category of simplicial objects over C will bedenoted by sC.

Of course, C = DGL0 (as well as DGA1) is itself a model category (as shownby Quillen in [42]), so in considering simplicial DGLs we can take advantage of theprocedure described by Dwyer, Kan, and Stover for generating a model categorystructure on sC. Essentially, this consists of:

• choosing a certain set Mαα∈A of “good” objects in C (correspondingto the projectives of homological algebra);

• creating simplicial objects ΣnMα for each n ∈ N and α ∈ A by placingMα in simplicial dimension n; and using these to

• define weak equivalences (which are detected by maps out of ΣnMα),and

• construct the cofibrant objects (corresponding to projective resolutions),and more generally all cofibrations.

See [19] for more details; as extended by Bousfield (who calls it a “resolutionmodel category”) in [14], this procedure works in great generality. In particular,it includes the model categories for universal algebras defined by Quillen in [41,II, §4], as well as the graded version (see, e.g., [12, §2.2]). The original example

44 DAVID BLANC

considered by Dwyer, Kan, and Stover was that of simplicial spaces; this can bereadily modified to cover simplicial rational spaces (equivalently, simplicial DGLsor DGAs).

3.3. Simplicial DGLs. In applying the above procedure to C = DGL0, wehave a natural candidate for the “good objects” – namely, the DGL spheresSn∞n=2 (which are just minimal models for Sn

Q∞n=2 – i.e., Sn = Sn

(x) is a free

graded Lie algebra on a single generator x in dimension n).This determines what we mean by a (simplicial) resolution of a given DGL

(L, ∂) – namely, a cofibrant simplicial DGL V , with each Vn homotopy equivalentto a coproduct

∐i∈In

Sni , such that the augmented simplicial DGL V → L isweakly contractible. A map ϕ : V → X of simplicial DGLs is a weak equivalenceif and only if after taking (internal) homology in each simplicial dimension, theresulting map of graded simplicial abelian groups ϕ∗ : H∗V → H∗X induces an

isomorphism Hn(ϕ∗) : Hn(H∗(V ))∼=−→ Hn(H∗(X)) for each n ≥ 0.

By [42, Thm. 2.1], we know that the model category DGL0 is equivalentto the category sLie1 of connected simplicial Lie algebras; similarly for bigradeddifferential Lie algebras and simplicial graded Lie algebras by [7, Prop. 2.9]. In fact,it turns out that there is also a one-to-one correspondence between filtered DGLsand (cofibrant) simplicial DGL resolutions (see [7, Prop. 5.13]), so the DGL defor-mation theory mentioned in §3.1 can be translated into the language of simplicialresolutions.

3.4. Moduli and realizations. One can analyze MP in terms of a real-ization problem: which distinct rational homotopy types of 1-connected topologicalspaces have the given graded Lie algebra P as their rational homotopy groups? Inview of the equivalences of categories noted above, we can replace the DGL bigradedmodel (B, ∂) for (P, 0) by a free simplicial resolution G → P of graded Liealgebras, and observe that if P itself can be realized by a rational space X (andso by a DGL (LX , ∂)), then G can be realized by a simplicial space (or DGL)V . Moreover, any weak equivalence ϕ : V → W of simplicial spaces (or DGLs)induces a weak equivalence ϕ# : π∗V ⊗Q → π∗W ⊗Q, and since π∗V ⊗Q = G isa resolution of P , this is just a choice of an isomorphism between two isomorphiccopies of P .

As we shall see below (§6), there is a “moduli space” associated to any modelcategory, and the various levels of classification which are a feature of moduli spacesin general (§1.1(i)) can be understood in terms of “structure-reducing” functorsbetween the corresponding model categories – here exemplified by π∗( ) ⊗ Q :T1 → grLie (extended to simplicial objects).

4. Moduli and higher homotopy operations

Ideally, we want not only an algebraic description of the components of MH

in terms of the cohomology of the graded algebra H , say, as in §2.10, but also ageometric interpretation of this description – with the potential for lifting to theintegral case.

Note that there is an obvious candidate for geometric invariants to distinguishbetween inequivalent rational spaces with the same cohomology – namely, Masseyproducts, and their higher order analogues (see [38, 33]). In fact, Halperin and

MODULI SPACES OF HOMOTOPY THEORY 45

Stasheff’s original motivation for their obstruction theory was to make precise afolk theorem stating that a space is formal if and only if all higher order Masseyproducts – or more generally, all higher-order cohomology operations – vanish.

One difficulty is in the very definition of higher cohomology operations (see,e.g., [37]). This can be bypassed to some extent by re-interpreting them in termsof differentials in a spectral sequence (cf. [27, §7]) – at the price of losing boththeir geometric content, and the ability to identify the same higher operation indifferent contexts.

Because their Eckmann-Hilton dual – namely, higher homotopy operations –is somewhat more intuitive than the cohomology version, we shall concentrate onthese. A precise definition requires some care; in the following sections, we presentthe version of [10]:

4.1. Definition. A lattice is a finite directed non-unital category Γ (that is,we omit the identity maps), equipped with two objects vinit = vinit(Γ) and vfin =vfin(Γ) with a unique map φmax : vinit → vfin, and for every w ∈ V := Obj Γ,there is at least one map from vinit to w, and at least one from w to vfin. Acomposable sequence of k arrows in Γ will be called a k-chain.

4.2. The W -construction. Given a lattice Γ, one can define a new categoryWΓ enriched over cubical sets, with the same set of objects V (cf. [13, III, §1]):

For u, v ∈ V , let Γn+1(u, v) be the set of (n + 1)-chains from u to v in Γ, and

WΓ(u, v) :=⊔

n≥0

Γn+1(u, v) × In/ ∼,

where I is the unit interval. Write

f1 t1 f2 · · · fn tnfn+1

for

〈ufn+1−−−→ vn · · · → v1

f1−→ v〉 × (tn, . . . , t1)

in Γn+1(u, v) × In; then the relation ∼ is generated by

f1 t1 f2 · · · fn tnfn+1 ∼ f1 t1 · · · ti−1 (fifi+1) ti+1 · · · tn

fn+1,

if ti = 0 for 1 ≤ i ≤ n, where (fifi+1) denotes fi composed with fi+1.The categorial composition in WΓ is given by the concatenation:

(f1 t1 · · · tlfl+1) (g1 u1 · · · uk

gk+1) := (f1 t1 · · · tlfl+1 1 g1 u1 · · · uk

gk+1).

We write P := WΓ(vinit, vf) .

4.3. Definition. The basis category bWΓ for a lattice Γ is defined to be thecubical subcategory of WΓ with the same objects, and with morphisms given bybWΓ(u, v) := WΓ(u, v) if (u, v) 6= (vinit, vfin), while

bWΓ(vinit, vfin) :=⋃

α β | β ∈ WΓ(vinit, w), α ∈ WΓ(w, vfin), vinit 6= w 6= vfin ,

so that bP := bWΓ(vinit, vfin) consists of all decomposable morphisms.

4.4. Fact ([10, Proposition 2.15]). For any lattice Γ, WΓ(vinit, vfin) is iso-morphic to the cone on its basis, with vertex corresponding to the unique maximal1-chain.

46 DAVID BLANC

We can use the W -construction to define a higher homotopy operations, asfollows:

4.5. Definition. Given A : Γ → hoT∗ for a lattice Γ as above, the cor-responding higher order homotopy operation is the subset 〈〈A〉〉 ⊂ [A(vinit) ⋊

bP , A(vfin)]ho T∗of the homotopy equivalence classes of maps

CA|bWΓ(vinit,vfin) : bWΓ(vinit, vfin) = bP −→ T∗(A(vinit), A(vfin))

induced by all possible continuous functors CA : bWΓ → T∗ such that π CA =A (ε|bWΓ), where the half-smash is defined X ⋊ K := (X × K)/(∗ × K) =X ∧ K+.

〈〈A〉〉 is said to vanish if it contains the homotopy class of a constant mapbP −→ T∗(A(vinit), A(vfin)).

Given a lattice Γ, a rectification of a functor A : Γ → ho T∗ is a functorF : Γ → T∗ with π F naturally isomorphic to A (where π : T∗ → hoT∗ is theobvious projection functor).

4.6. Proposition ([10, Thm. 3.8]). A has a rectification if and only if thehomotopy operation 〈〈A〉〉 vanishes (and in particular, is defined)

With this at hand, we can now reformulate the deformation theory for thebigraded DGL model for (P, 0) in the following form:

4.7. Theorem ([7, Thm. 7.14]). For each connected graded Lie algebra P offinite type, there is a tree TP , with each node α indexed by a cofibrant DGL Lα

such that H∗Lα ∼= P , starting with L0 ≃ (P, 0) at the root 0. For each node

α there is an integer nα > 0 such that Lα agrees with its successors in degrees≤ nα, with n0 = 1, so that the sequential colimit L(∞) = colimk L(k) along anybranch is well defined, and for any rational homotopy type in MP there existssuch a tree. Furthermore, for each two immediate successors β, γ of a node α,there is a higher homotopy operation 〈〈β, γ〉〉 ⊆ [Sn, L] which vanishes for L = Lβ,but not for L = Lγ (or conversely).

Thus we have reformulated the deformation of the bigraded model as an ob-struction theory for distinguishing between the various realizations of a given ratio-nal graded Lie algebra P in terms of higher homotopy operations. This can also bedone integrally (cf. [4]), and one could identify these operations with appropriatecohomology classes in the dual version of the Halperin-Stasheff obstructions (see[5]).

4.8. Example of a DGL deformation. We shall not explain how the higheroperations 〈〈β, γ〉〉 are defined, since the construction is rather technical; see [7,§4] for the details. Instead, the reader might find the following example instructive:

Consider the graded Lie algebra P = L〈a1, b1, c2〉/I, where L〈T∗〉 is the freegraded Lie algebra on a graded set of generators T∗, (with the subscripts indicatingthe dimension), and I is here the Lie ideal generated by [a, a] and [[c, a], [b, a]].

MODULI SPACES OF HOMOTOPY THEORY 47

Step I: The minimal model for the coformal DGL (P, 0) ∈ DGL0 is (B, ∂B),where B in dimensions ≤ 7 is L〈a1, b1, c2, x3, y5, w6, z7〉, with

∂B(x) = [a, a],

∂B(y) = 3[x, a],

∂B(w) = [[c, a], [b, a]],

∂B(z) = 4[y, a] + 3[x, x].

The bigraded model A∗∗ is obtained from B by introducing an additional (homo-logical) grading: a, b ∈ A0,1, c ∈ A0,2, x ∈ A1,3, w ∈ A1,6, y ∈ A2,5, z ∈ A3,7,and so on.

Step II: The free simplicial DGL resolution C• → L = (L, 0) may be describedin homological dimensions ≤ 3 as follows:

(1) C(0)0 is the DGL coproduct of

S1(a(0))

= S1(〈a〉),

S1(b(0))

= S1(〈b〉),

S2(c(0))

= S2(〈c〉).

(2) C(0)1 = S2

(x(0))∐ S6

(w(0)), where

x(0) = 〈[〈a〉, 〈a〉]〉,

w(0) = 〈[[〈c〉, 〈a〉], [〈b〉, 〈a〉]]〉.

(3) C(0)2 consists of S5

(y(0)), where

y(0) = 〈3[〈[〈a〉, 〈a〉]〉, 〈〈a〉〉]〉.

(4) C(0)3 consists of S

6(z(0))

, where

z(0) = 〈4[〈3[〈[〈a〉, 〈a〉]〉, 〈〈a〉〉]〉, 〈〈〈a〉〉〉] + 6[〈[〈〈a〉〉, 〈〈a〉〉]〉, 〈〈[〈a〉, 〈a〉]〉〉]〉.

In analogy with the DGL sphere of §3.3, the DGL n-disk Dn(x) is the free

graded Lie algebra on two generators: x in dimension n and ∂(x) in dimension

n − 1. With this notation, for C(1)• we need in addition:

(1) D3(x(1))

→ C(1)0 with

∂W (x(1)) = d1(x(0)) = 〈[a, a]〉.

(2) D6(y(1))

→ C(1)1 with

y(1) = 〈3[〈x〉, 〈a〉]〉

∂W (y(1)) = d2(y(0)) = 〈3[〈[a, a]〉, 〈a〉]〉.

(3) D7(z(1))

→ C(1)2 with

z(1) = 〈4[〈3[〈x〉, 〈a〉]〉, 〈〈a〉〉] + 6[〈[〈a〉, 〈a〉]〉, 〈〈x〉〉]〉

∂W (z(1)) = d3(z(0)) = 〈4[〈3[〈[a, a]〉, 〈a〉]〉, 〈〈a〉〉] + 6[〈[〈a〉, 〈a〉]〉, 〈〈[a, a]〉〉]〉.

For C(2)• we need in addition

48 DAVID BLANC

(1) D7(y(2))

→ C(2)0 with

∂W (y(2)) = d1(y(1)) = 〈3[x, a]〉.

(2) D8(z(2))

→ C(2)1 with

z(2) = 〈4[〈y〉, 〈a〉] + 3[〈x〉, 〈x〉]〉

∂W (z(2)) = d2(z(1)) = 〈4[〈3[x, a]〉, 〈a〉] + 6[〈[a, a]〉, 〈x〉]〉.

For C(3)• we must add D9

(z(3))→ C

(3)0 with

∂W (z(1)) = d1(z(2)) = 〈4[y, a] + 3[x, x]〉.

Step III: So far, we have constructed the DGL simplicial resolution of thecoformal DGL (B, ∂B) ≃ (P, 0). Now we consider a different DGL having the samehomology (i.e., a non-weakly equivalent topological space with the same rationalhomotopy groups):

Consider the DGL (B′, ∂B′) where B′ := L〈a1, b1, c2, x3, y5, z7, . . .〉, is afree Lie algebra with ∂B′(x) = [a, a], ∂B′(y) = 3[x, a] − [[b, a], c], ∂B′(z) =4[y, a] + 3[x, x], and so on.

Here P ∼= H∗(B) = L〈a1, b1, c2〉/〈[a, a], [[c, a], [b, a]]〉, so the bigraded modelfor (P, 0) is (A∗∗, ∂B) of Step II above, and the filtered model is obtained from itby setting D(y) = 3[x, a]− [[b, a], c] and D(z) = 4[y, a] + 3[x, x]− 4w + 2[[x, b], c].

The corresponding free simplicial DGL resolution is obtained from C• of StepII by making the following changes:

(1) Set

y(1) := 〈3[〈x〉, 〈a〉] − [[〈b〉, 〈a〉], 〈c〉]〉,

with

∂W (y(1)) = d2(y(0)) = 〈3[〈[a, a]〉, 〈a〉]〉

as before (but now ∂W (y(2)) = 〈3[x, a] − [[b, a], c]〉, of course).(2) Set

z(1) := 〈4[〈3[〈x〉, 〈a〉]〉, 〈〈a〉〉] + 6[〈[〈a〉, 〈a〉]〉, 〈〈x〉〉]

− 4〈[[〈c〉, 〈a〉], [〈b〉, 〈a〉]]〉 + 2[[〈[〈a〉, 〈a〉]〉, 〈〈b〉〉], 〈〈c〉〉]〉

(with ∂W (z(1)) unchanged).(3) Set

z(2) := 〈4[〈y〉, 〈a〉] + 6[〈x〉, 〈x〉] − 4〈w〉 + 2[[〈x〉, 〈b〉], 〈c〉]〉,

with

∂W (z(2)) = d3(z(1)) = 〈4[〈3[x, a] − [[b, a], c]〉, 〈a〉]

+ 6[〈[a, a]〉, 〈x〉] − 4〈[[c, a], [b, a]]〉 + 2[[〈[a, a]〉, 〈b〉], 〈c〉]〉

(4) Finally,

∂W (z(3)) = 〈4[y, a] + 3[x, x] − 4w + 2[[x, b], c]〉.

MODULI SPACES OF HOMOTOPY THEORY 49

Step IV: In order to define the higher homotopy operations which distinguishB′ from B, observe that the required half-smashed are defined as follows:

S2(x) ⋊ D[1] = (L〈X∗〉, ∂

′),

whereX2 = (x, (d0)), (x, (d1))

X3 = (x, (Id)),

and∂′(x, (Id)) = (x, (d0)) − (x, (d0)).

Similarly,S

3(y) ⋊ D[2] = (L〈Y∗〉, ∂

′),

whereY3 = (y, (d0d1)), (y, (d0d2)), (y, (d1d2)),

Y4 = (y, (d0)), (y, (d1)), (y, (d2)),

Y5 = (y, (Id)),

with∂′(y, (Id)) = −(y, (d0)) + (y, (d1)) − (y, (d2)),

∂′(y, (d0)) = −(y, (d0d1)) + (y, (d0d2)),

∂′(y, (d1)) = −(y, (d0d1)) + (y, (d1d2)),

∂′(y, (d2)) = −(y, (d0d2)) + (y, (d1d2)).

Step V: For the DGL B′ of Step III, with C• as in Step II, we defineh0 = f0 : C0,∗ → B′ by setting f0(〈a〉) = a, f0(〈b〉) = b, f0(〈c〉) = c, and f0 = 0for all other disks in C0,∗ (so for example f0(〈[a, a]〉) = 0).

Thus on S2(x(0))

⋊ D[1] we have

h1(x(0), (d0)) = [a, a],

h1(x(0), (d1)) = 0,

(see [7, Def. 4.7]), so we must choose h1(x(0), (Id)) = x ∈ B′

3.Now on S

3(y(0))

⋊ D[2] we have

h2(y(0), (d0d1)) = h2(y

(0), (d0d2)) = h2(y(0), (d1d2)) = 0,

h2(y(0), (d0)) = (h1 d0)(y

(0), (d0)) = 3[x, a],

h2(y(0), (d1)) = h2(y

(0), (d2)) = 0.

Thus in Definition 4.5 we shall be interested in a lattice Γ corresponding tothe face maps of a simplicial object, with bWΓ ∼= ∂D[2]. We have just defined acontinuous functor CA : bWΓ → T∗ for C•, as required there, and the resultingsecondary operation is 〈〈2, y(0)〉〉 = 〈3[x, a]〉] ⊆ H4B

′ (remember that the ho-mology of a DGL corresponds to the rational homotopy groups of the associatedspace). Since 3[x, a] does not bound in B′, 〈〈2〉〉 does not vanish, and we havefound an obstruction to the coformality of B′.

Unfortunately, DGL higher homotopy operations such as 〈〈2, y(0)〉〉 are un-satisfactory, in as much as one cannot translate them canonically into integralhomotopy operations. One way to avoid this difficulty is to use more general non-associative algebras, rather than Lie algebras, as our basic models. Thus, considerthe category DGN of non-associative differential graded algebras. A DGN whosehomology happens to be a graded Lie algebra will be called a Jacobi algebra. Every

50 DAVID BLANC

DGL has a free Jacobi model, and these can be used to resolve any DGL in sDGN(see [7, §7]).

Step VI: Consider a Jacobi model G for the coformal DGL P of Step I. Weconstruct the minimal Jacobi resolution J• → G corresponding to the bigradedmodel for L (embedded in the canonical Jacobi resolution U• → G) by modifyingthe free resolution C• → L of Step II, as follows:

For each n ≥ 0 define the Jacobi algebras J(0)n (n = 0, 1, . . . ) to be the

coproducts

J(0)0 = S1

(〈a〉) ∐ S1(〈b〉) ∐ S2

(〈c〉)

J(0)1 = S2

(x(0))∐ S6

(w(0))

J(0)2 = S5

(y(0))

J(0)3 = S6

(z(0)),

and so on.The face maps di (i = 0, 1, . . . , n) are defined as follows: write 〈x〉 ∈ F (B)

for the generator corresponding to an element x ∈ B′∗, then recursively a typical

DGL generator for Wn = Wn,∗ (in the canonical Stover resolution W•(B), definedin [7, §5.3]) is 〈α〉, for α ∈ Wn−1, so an element of Wn is a sum of iterated Lieproducts of elements of B′

∗, arranged within n+1 nested pairs of brackets 〈〈· · ·〉〉.With this notation, the i-th face map of W• is “omit i-th pair of brackets”, andthe j-th degeneracy map is “repeat j-th pair of brackets”. We assume the bracketoperation 〈−〉 is linear – i.e., that 〈αx + βy〉 = α〈x〉 + β〈y〉 for α, β ∈ Q andx, y ∈ B.

1. For y(0) = 〈3[〈[〈a〉, 〈a〉]〉, 〈〈a〉〉]〉 ∈ J(0)2 3, we have

d0(y(0)) = 3[〈[〈a〉, 〈a〉]〉, 〈〈a〉〉] = 3[x(0), a(0)],

d2(y(0)) = 〈3[〈[a, a]〉, 〈a〉]〉,

d1(y(0)) = 〈3[[〈a〉, 〈a〉], 〈a〉]〉.

This no longer vanishes as in Step II, since the Jacobi identity does not hold in

DGN , but we have an element y(1;1) := 〈λ3(〈a〉 ⊗ 〈a〉 ⊗ 〈a〉)〉 ∈ J(1)1 4 (in the

notation of [7, Example 7.8], with ∂(y(1;1)) = d1(y(0)).

On the other hand, we also have an element y(1;2) := 〈3[〈x〉, 〈a〉]〉 ∈ J(1)1 4

(which we denoted simply by y(1) in Step II, with ∂(y(1;2)) = d2(y(0)). The

simplicial identity d1d2 = d1d1 implies that

d1(y(1;2)) − d1(y

(1;1)) = 〈3[x, a] − λ3(a ⊗ a ⊗ a)〉

is a ∂J -cycle, so we have y(2;1,2) ∈ J(2)0 5 with

∂J(y(2;1,2)) = 〈3[x, a] − λ3(a ⊗ a ⊗ a)〉.

2. For z(0) = 〈4[〈3[〈[〈a〉, 〈a〉]〉, 〈〈a〉〉]〉, 〈〈〈a〉〉〉] + 6[〈[〈〈a〉〉, 〈〈a〉〉]〉, 〈〈[〈a〉, 〈a〉]〉〉]〉 in

J(0)3 4:

MODULI SPACES OF HOMOTOPY THEORY 51

(a) z(1;1) := 〈λ3(x(0) ⊗ 〈〈a〉〉 ⊗ 〈〈a〉〉)〉, with

d1(z(0)) = 〈6(2[[x(0), 〈〈a〉〉], 〈〈a〉〉], +[[〈〈a〉〉, 〈〈a〉〉], x(0)])〉 = ∂J (z(1;1)),

(b) z(1;2) := 〈4[λ3(〈a〉 ⊗ 〈a〉 ⊗ 〈a〉), 〈〈a〉〉]〉, with

d2(z(0)) = 〈4[[[a(0), a(0)], a(0)], 〈〈a〉〉]〉 = ∂J(z(1;2)) since [x(0), x(0)] = 0).

(c) z(1;3) = 〈4[〈3[〈[a, a]〉, 〈a〉]〉, 〈〈a〉〉] + 6[〈[〈a〉, 〈a〉]〉, 〈〈[a, a]〉〉]〉 ∈ J(1)2 5, with

d3(z(0)) = ∂J (z(1;3)).

3. Next, in J(2)• :

(a) The simplicial identity d1d1 = d1d2 implies that

d1(z(1;1)) − d1(z

(1;2)) = 〈4[λ3(〈a〉 ⊗ 〈a〉 ⊗ 〈a〉), 〈a〉]

− 6λ3([〈a〉, 〈a〉] ⊗ 〈a〉 ⊗ 〈a〉)〉

is a ∂J -cycle – and indeed we have

z(2;1,2) := 〈λ4(〈a〉 ⊗ 〈a〉 ⊗ 〈a〉 ⊗ 〈a〉)〉 ∈ J(2)2 6

with∂(z(2;1,2)) = d1(z

(1;1)) − d1(z(1;2)).

(b) Similarly,

d1(z(1;3)) − d2(z

(1;1)) = 〈12[[〈x〉, 〈a〉], 〈a〉] + 6[[〈a〉, 〈a〉], 〈x〉]

− 6λ3(〈[a, a]〉 ⊗ 〈a〉 ⊗ 〈a〉)〉

is a ∂J -cycle, so we have

z(2;1,3) := 〈6λ3(〈x〉 ⊗ 〈a〉 ⊗ 〈a〉)〉 ∈ J(2)2 6

with∂(z(2;1,3)) = d1(z

(1;3)) − d2(z(1;1)).

(c) d2(z(1;3)) − d2(z

(1;2)) = 〈4[〈3[x, a]〉, 〈a〉] + 6[〈[a, a]〉, 〈x〉]

− 4[〈λ3(a ⊗ a ⊗ a)〉, 〈a〉]〉

is a ∂J -cycle, hit by

z(2;2,3) := 〈4[〈y〉, 〈a〉] + 3[〈x〉, 〈x〉]〉 ∈ J(2)2 6.

4. Finally, we have

(a) For d1z(2;1,2) = 〈λ4(a ⊗ a ⊗ a ⊗ a)〉 we have

∂J(d1z(2;1,2)) = d1d2(z

(1;2)) − d1d2(z(1;1))

= 〈4[λ3(a ⊗ a ⊗ a), a] − 6λ3([a, a] ⊗ a ⊗ a)〉

(b) For d1z(2;1,3) = 〈6λ3(x ⊗ a ⊗ a)〉 we have

∂J (d1z(2;1,3)) = d1d2(z

(1;1)) − d1d1(z(1;3))

= 〈6λ3([a, a] ⊗ a ⊗ a) − 12[[x, a], a]− 6[[a, a], x]〉

(c) For d1z(2;2,3) = 〈4[y, a] + 3[x, a]〉 we have

∂J(d1z(2;2,3)) = d1d2(z

(1;2)) − d1d1(z(1;3))

= 〈4[λ3(a ⊗ a ⊗ a), a] − 12[[x, a], a] − 6[[a, a], x]〉,

52 DAVID BLANC

So there is an element z(3;1,2,3) ∈ J(3)0 7 with

∂J(z(3;1,2,3)) = 〈λ4(a ⊗ a ⊗ a ⊗ a) − 6λ3(x ⊗ a ⊗ a) + 4[y, a] + 3[x, a]〉.

This defines the second-order homotopy operation we are interested in (which alsohas an integral version).

5. Refined moduli spaces

Since neither the cohomolgy algebra nor the homotopy Lie algebra of a rationalspace determine the other, we can refine the coarse partition of the rational modulispace M by specifying both:

5.1. Definition. For each 1-connected graded-commutative algebra H andgraded Lie algebra P over Q, let MH,P denote the collection of all simply-connected rational homotopy types of spaces X with H∗(X ; Q) ∼= H and π∗(X)⊗Q ∼= P .

Note that we cannot simply identify this as a quotient variety with MH ∩MP

in any natural way, since points in MH are represented by filtered DGA models,while those in MP are represented by filtered DGA models.

The set MH,P may, of course, be empty; but Lemaire and Sigrist have shownthat it can also be infinite (see [34, §3]). In order to analyze this refined modulispace, we shall need one additional ingredient.

5.2. The Quillen equivalences. The fact that hoDGL0 and ho DGA1 areequivalent to each other, leads us to expect a direct algebraic relationship betweenthe corresponding model categories, which in fact exists, and may be described asfollows:

For simplicity, restrict attention to simply-connected spaces of finite type. Wemay therefore assume that Ai

X is finite dimensional for each i ≥ 0 (and ofcourse A0

X = Q, A1X = 0). Taking the vector space dual of A∗

X , we obtain a 1-connected differential graded-cocommutative coalgebra (CX

∗ , δ), whose homologyis H∗(X ; Q), with the usual coalgebra structure (dual to the ring structure incohomology). Let DGC denote the category of such coalgebras.

Quillen defined a pair of adjoint functors

DGL0C

LDGC

as follows:

I: Given a DGC (C∗, d), let L(C∗) := (Prim (ΩC∗), ∂) denote the graded Liealgebra of primitives in the cobar construction of C∗, constructed as follows:

If C∗∼= Q ⊕ C∗ (where C∗ = C≥2, in our case), and Σ−1V is the graded

vector space V shifted downwards (so that (Σ−1V )i := Vi+1, with σ−1v ↔ v),let ΩC∗ := T (Σ−1C∗) denote the tensor algebra on Σ−1C∗ with

∂(σ−1x) := −σ−1(dx) +1

2

∑

i

(−1)|c′

i|[σ−1c′i, σ−1c′′i ] ,

where ∆c :=∑

i c′i ⊗ c′′i is the (reduced) comultiplication in C∗, and [ , ] is thecommutator in T (Σ−1C∗).

MODULI SPACES OF HOMOTOPY THEORY 53

II: Given a DGL (L∗, ∂), let C(L∗, d) be the DGC (Λ(ΣL∗), d), where ΣL∗ isL∗ shifted upwards, ΛV denotes the cofree graded coalgebra cogenerated by thegraded vector space V , and the coderivation ∂ is defined by d = d′ − d′′, where

d′(σx1 ∧ . . . ∧ σxn) :=n∑

i=1

(−1)i+P

j<i |xj |σx1 ∧ . . . ∧ σ∂xi ∧ . . . ∧ σxn),

d′′(σx1 ∧ . . . ∧ σxn) :=∑

1≤i<j≤n

(−1)|xi|(±)σ[xi, xj ] ∧ σx1 . . . σxi . . . σxj . . . σxn) ,

(5.1)

and the sign (±) is determined by

σx1 ∧ . . . ∧ σxn = (±)σxi ∧ σxj ∧ σx1 ∧ . . . σxi . . . σxj . . . ∧ σxn) .

See [42, App. B] or [25, §22] for the details.

5.3. Joint deformations. This suggests the following approach to the refinedmoduli space for a graded Lie algebra P and graded cocommutative coalgebra H ,where we assume H and P are both of finite type, H is n-connected and P is(n − 1)-connected (n ≥ 1), and Hn+1

∼= Pn:Let (B, ∂) be the bigraded model for (P, 0), and (C, d) = (C∗B, d′ −

d′′) the corresponding coalgebra (§5.2 II). Note that (C, d) is cofibrant, but notnecessarily minimal. As we deform (B, ∂) to produce all of MP , we change

only d′ in the coalgebra model, obtaining (B, ∂), say. At the same time, we

have the deformations (A, d) say, of the formal DGC (H, 0); and for each

pair 〈B, A〉, we have the variety V〈B,A〉 of all DGC maps φ : (B, ∂) → (A, d)

between them. Since both are cofree, these are determined by maps of gradedvector spaces. The condition that φ be a quasi-isomorphism translates into a seriesof rank conditions, so that we get a semi-algebraic set parametrizing all such pairswhich are equipped with a quasi-isomorphism between them – and in particular,the requisite homology H and homotopy groups P .

6. Nerves and moduli spaces

Since ordinary (integral) homotopy types do not have any known differentialgraded models, there is no hope of generalizing the deformation approach of sections2-5 to cover them, too. However, in [18], Dwyer and Kan suggested an approachto such “moduli problems” based on the concept of nerves, which has proved usefulconceptually in a number of contexts.

6.1. Definition. The nerve NC of a small category C is a simplicial set whosek-simplices are the sequences of k composable arrows in C, with di=“delete i-thobject and compose”, sj=“insert identity arrow after j-th object”. The geometricrealization of the nerve is called the classifying space of C, written BC := |NC|.

The nerve was originally defined by Segal in [48], based on ideas of Grothen-dieck; Quillen, in [43] helped clarify the close connection between nerves and ho-motopy theory, as evinced in the following properties:

(1) The functor N : Cat → S takes natural transformations to homotopies(cf. [43, §2, Prop. 2]), so that

54 DAVID BLANC

(2) if a functor F has a left or right adjoint, then NF is a homotopyequivalence; more general conditions are provided by [43, §2, Thm. A]).If C has an initial or final object, then NC ≃ ∗.

(3) The nerve of a functor is not often a fibration; conditions when NF fitsinto a quasi-fibration sequence are provided by [43, §2, Thm. B].

(4) Similarly, NC is not usually a Kan complex, unless C is a groupoid (cf.[26, I, Lemma 3.5]).

6.2. Definition. A classification complex in a model category C (see [18, §2.1])is the nerve of some subcategory D of C, all of whose maps are weak equivalences,and which includes all weak equivalences whose source or target is in D. Thecategory D is only required to be homotopically small (cf. [16, §2.2]) – that is,ND has a set of components, and its homotopy groups (at each vertex) are small.

This construction has two main properies:

(a) The components of ND are in one-to-one correspondence with the weakhomotopy types of D in C.

(b) The component of ND corresponding to an object X ∈ C is weaklyhomotopy equivalent to the classifying space of the monoid hautX ofself weak equivalences of X (cf. [18, Prop. 2.3]).

6.3. Remark. Taking the model category C := T0 of connected pointedspaces, with D = WC the subcategory of all weak homotopy equivalences inC, MC := NWC is the candidate suggested by the Dwyer-Kan approach for themoduli space of pointed homotopy types. There is also an unpointed version, ofcourse; on the other hand, allowing for non-connected spaces merely complicatesthe combinatorics of M, without adding any new information.

One might ask in what sense this qualifies as a moduli space (aside from hav-ing the right components). Even though the analogy with the classical examplesmentioned in the introduction is not clear-cut, note that MT0 as defined usingthe nerve (which is natural choice for a space to associate to a category) is related

to MT Q1 of §2.5, and the latter does exhibit many of the attributes listed in

§1.1. Moreover, as we shall see in the next section, this construction can be used tointerpret the obstruction theory of §2.10, as well as its integral analogue, and alsoto describe M as the limit of a tower of fibrations, which give increasingly accurateapproximations to M.

Finally, the monoid hautX of self equivalences corresponds to the mappingclass group of self diffeomorphisms of a surface (Sec. 1), whose classifying space isclosely related (or even homotopy equivalent) to the moduli space.

6.4. The model categories. In order to provide a uniform treatment indifferent model categories, and in particular to allow for the comparisons mentionedin §3.4, it is useful to consider a resolution model category structure (§3.2) on acategory C = sE of simplicial objects over another category E . In fact, one can dothis at two different levels, and in some sense the comparison between these is theheart of this approach:

I. The topological level – where E can take several forms:

(a) The category T0 of connected pointed spaces.

MODULI SPACES OF HOMOTOPY THEORY 55

(b) The subcategory T Q0 of pointed connected spaces having rational uni-

versal covers (but arbitrary fundamental group).Note that one can approximate any pointed connected space X ∈ T0

by its fibrewise rationalization X ′Q (cf. [15, I, 8.2]), which lies in T Q

0 : if

X → X → B(π1X) is the universal covering space fibration for X , then

X ′Q fits into a functorial fibration sequence XQ → X ′

Q → B(π1X), in

which XQ is the usual rationalization of the universal cover. However,algebraic DGL, DGA, or DGC models do not generally extend to this case(unless π1X is finite and acts nilpotently on the higher groups – see,e.g., [54]).

(c) The subcategory T Q1 of 1-connected pointed rational spaces (in T Q

0 ).This can be replaced by the Quillen equivalent model categories DGL0

or DGA1.(d) Other variants are possible – for instance, we could consider functor

categories over E – that is, diagrams D → E for a fixed small categoryD (see [9]).

II. The algebraic level – where E is correspondingly:

(a) The category Π-Alg of Π-algebras – that is, positively graded groupsG∗, abelian in dimensions ≥ 2, equipped with an action of the primaryhomotopy operations (Whitehead products, compositions, and G1-action)satisfying the usual identities (see [2] for a more explicit definition).

(b) The subcategory Π-AlgQ of rational Π-algebras, which are (positively)graded Lie algebras equipped with a “fundamental group action” of anarbitrary group π, satisfying the usual identities (see [30]).

(c) When π = 0, we obtain the subcategory Π-AlgQ1∼= grLie of simply-

connected rational Π-algebras, which are just graded Lie algebras overQ.

(d) There is also a concept of Π-algebras for arbitrary diagrams (again, see[9]).

7. Approximating classification complexes

The classification complex of a model category C, as defined in §6.2, may appearto be a somewhat artificial marriage of a traditional moduli space, whose set ofcomponents correspond to the (weak) homotopy types of C, and the individualcomponents, which are classifying spaces of the form B hautX . It is not clear atfirst glance why such complexes might be useful. To understand this, we show howMC can be approximated by a tower of fibrations, in a way that elucidates theobstruction theory of §2.10:

7.1. Postnikov systems and Eilenberg-Mac Lane objects. Recall from[17, §1.2] that functorial Postnikov towers

X → . . . → PnX → Pn−1X → . . . → P0X

may be defined in categories of the form C = sE using the matching space construc-tion of [15, X,§4.5]. By considering the fibers of successive Postnikov sections, wecan define the analogue of homotopy groups in each such category, and find that

56 DAVID BLANC

they are corepresented, as one might expect, by appropriate suspensions of the“good objects” in E . For both E = T0 and E = Π-Alg, we find that the natural“homotopy groups” πnX of any X ∈ sE (the bigraded groups of [20]) take valuesin Π-Alg, with the obvious modifications for the rational variants.

Note that for a simplicial space X ∈ sT0, applying π∗ in each simplicialdimension yields a simplicial Π-algebra π∗X , and the two sequences of Π-algebras(πnX)∞n=0 and (πnπ∗X)∞n=0 fit into a “spiral long exact sequence”:(7.2)

. . . πn+1π∗X∂⋆

n+1−−−→ Ωπn−1X

sn−→ πnXhn−−→ πnπ∗X

∂⋆n−→ · · · π0X

h0−→ π0π∗X → 0

(see [20, 8.1]), in which each term is not only a Π-algebra, but a module overπ0X ∼= π0π∗X under a “fundamental group action”, for n ≥ 1. Here ΩΛdenotes the abelian Π-algebra obtained from a Π-algebra Λ by re-indexing andsuspending the operations (so that in particular Ωπ∗X ∼= π∗ΩX for any space X).

Furthermore, one can construct classifying objects BΛ ∈ sT0 for any Π-algebra Λ (with π0BΛ = Λ and πiBΛ = 0 otherwise); Eilenberg-Mac Laneobjects B(Λ, n); and twisted Eilenberg-Mac Lane objects BΛ(M, n) for anyΠ-algebra Λ, Λ-module M , and n ≥ 1, with

πiBΛ(M, n) ∼=

Λ if i = 0

M (as a Λ-module) if i = n

0 otherwise.

This is true more generally for resolution model categories, under reasonableassumptions. To simplify the notation we denote by KΛ the analogous simplicialΠ-algebra (with πnKΛ

∼= Λ for n = 0, and 0 otherwise), and similarly theEilenberg-Mac Lane objects K(Λ, n), and twisted Eilenberg-Mac Lane objectsKΛ(M, n) in sΠ-Alg. We shall use boldface in general to indicate constructionsin sT0, as opposed to sΠ-Alg.

Finally, one can define natural k-invariants for the Postinkov system of a simpli-cial object X ∈ sE , as in [8, Prop. 6.4], and these take values in appropriate Andre-Quillen cohomolgy groups (represented by the twisted Eilenberg-Mac Lane objects).These groups are denoted respectively by Hn(X/Λ; M) := [X,BΛ(M, n)]BΛ, fora simplicial space X equipped with a map to BΛ, and Hn(G/Λ; M) :=[G, KΛ(M, n)]KΛ , for a simplicial Π-algebra G equipped with a map to KΛ.

It turns out that for any simplicial space X there is a natural isomorphism

(7.3) Hn(X/Λ; M)∼=−→ Hn(π∗X/Λ; M)

for every n ≥ 1 (cf. [8, Prop. 8.7]).

7.4. Relating classification complexes. We shall now show how when E =T0 – and more generally – the classification complex MT0 can be exhibitedas the homotopy limit of a tower of fibrations, where the successive fibers havea cohomological description showing the relationship between π0MT0 and thehigher homotopy groups. The tower in question is constructed essentially by takingsuccessive Postnikov sections. The idea is an old one, and is useful even in analyzingthe self-equivalences of a single space X (see, for example, [55]).

7.5. Definition. For a given Π-algebra Λ, we denote by D(Λ) = DsT0(Λ) thecategory of simplicial spaces X ∈ sT0 such that π∗X ≃ BΛ (in sΠ-Alg) (that

MODULI SPACES OF HOMOTOPY THEORY 57

is, πnπ∗X ∼= Λ for n = 0, and πnπ∗X − 0 otherwise). The nerve of DsT0(Λ)will be denoted by MΛ.

The “pointed” version is the nerve of the category R(Λ) of pairs (X, ρ),where X ∈ sT0 and ρ : BΛ → π∗X is a specified weak equivalence in sΠ-Alg(again with weak equivalences as morphisms).

Although MΛ is the more natural object of interest in our context, we actuallystudy R(Λ). As noted in [8, §1.1], there is a fibration sequence

(7.6) NR(Λ) → MΛ → B Aut(Λ) ,

where Aut(Λ) is the group of automorphisms of the Π-algebra Λ.

7.7. Definition. For each n ≥ 1, let Rn(Λ) denote the category of n-Postnikov sections under KΛ – that is, the objects of Rn(Λ) are pairs (X〈n〉, ρ),where X〈n〉 ∈ sT0 is a simplicial space such that PnX〈n〉 ≃ X〈n〉 (Postnikovsections in sT0), and ρ : PnKΛ → Pnπ∗X〈n〉 is a weak equivalence. Themorphisms of Rn(Λ) are weak equivalences of simplicial spaces compatible withthe maps ρ up to weak equivalence of simplicial Π-algebras.

7.8. A tower of realization spaces. Given Λ, the Postnikov section functorsPn : sT0 → sT0 of §7.1 induce compatible functors Φn : R(Λ) → Rn(Λ) andFn : Rn+1(Λ) → Rn(Λ); and as in [8, Thm. 9.4] and [18, Thm. 3.4], these in turninduce a weak equivalence

(7.9) NR(Λ) → holimn NRn(Λ).

Combining (7.6) and (7.9), we may try to obtain information about the space ofrealizations MΛ by studying the successive stages in the tower

(7.10) . . .NRn+1(Λ)NFn−−−→ NRn(Λ)

NFn−1−−−−−→ . . . → NR1(Λ).

8. Analyzing the tower

The first step in analyzing the tower (7.10) is to understand when the successivefibers are non-empty, and if so, to count their components. In the rational case, too,our main task was identifying the components of the space of rational homotopytypes, and a partial ordering on the components was induced by the successivedeformations. The problem of empty fibers did not arise there, since all DGLs (orDGAs) are realizable by rational spaces. However, the fact that we have orderedthe successive choices in a tower, rather than a tree as in Theorem 4.7, suggeststhat we can describe them by means of an obstruction theory, as follows:

Assume given a point (X〈n〉, ρ) in NRn(Λ), so that X〈n〉 ∈ sT0 is asimplicial space which is an n-Postnikov stage (for some putative simplicial spaceY realizing the given Π-algebra Λ), and ρ : KΛ → Pnπ∗X〈n〉 is a choice of aweak equivalence. We can then reinterpret [8, Prop. 9.11] as saying:

8.1. Proposition. X〈n〉 ∈ Rn(Λ) extends to an (n + 1)-Postnikov stage inRn+1(Λ) if an only if the n + 1-st k-invariant for the simplicial Π-algebra π∗Xvanishes in Hn+3(Λ; Ωn+1Λ).

58 DAVID BLANC

The spiral long exact sequence (7.2) actually determines the homotopy “groups”of the simplicial Π-algebra π∗X〈n〉 completely:

(8.2) πkπ∗X〈n〉 ∼=

Λ for k = 0,

Ωn+1Λ for k = n + 2

0 otherwise.

However, (8.2) in itself does not imply that π∗X〈n〉 is an twisted Eilenberg-Mac Lane object KΛ(Ωn+1Λ, n + 2) in sΠ-Alg – for that to happen, the mapπ∗X〈n〉 → KΛ must have a section (whose existence is equivalent to the vanishingof the k-invariant in Proposition 8.1).

8.3. Remark. It may help to understand why if one considers a simplicialgroup K ∈ G (as a model for a connected topological space). The fundamentalgroup then appears as Γ = π0K, the set of path components of K (all homotopyequivalent to each other) – which happens to have a group structure. WhenK = KΓ(M, n) = K(M, n) ⋉ Γ is a twisted Eilenberg-Mac Lane object, the choiceof the section s : BΓ → K is what distinguishes K from a disjoint union (ofcardinality |Γ|) of copies of ordinary Eilenberg-Mac Lane objects K(M, n) –that is, L :=

∐|Γ| K(M, n). Even though we can put a group structure on π0L

so as to make it abstractly isomorphic to Γ, we will not get the right action of Γon M (which, in the case of K, appears in the usual way by conjugation with anyrepresentative of γ ∈ Γ = π0K).

8.4. Distinguishing between liftings. Since the weak homotopy types ofthe realizations of a given Π-algebra Λ (that is, components of MT0

Λ ) are in one-to-one correspondence with the weak homotopy types of the simplicial spaces Yrealizing KΛ in sT0 (if any), we should be able to describe them inductively interms of the simplicial-space k-invariants of Y , and thus of the successive Postnikovapproximations X〈n〉.

So, one naturally expects that there will be a correspondence between thesections sn and the k-invariants kn in sT0. However, by [8, Prop 9.11](using the identification of (7.3)) we know that the map φ : KΛ(Ωn+1Λ, n + 2) →KΛ(Ωn+1Λ, n + 2) corresponding to kn must be a weak equivalence, in our case– that is, it is within the indeterminacy of the k-invariants, which is certainlycontained in AutΛ(Ωn+1Λ).

Thus, the relevant information for constructing successive stages in a Postnikovtower for the simplicial space Y with π∗Y ≃ KΛ (if it exists), aside from the Π-algebra Λ, is not its k-invariants, nor the k-invariants for the simplicial Π-algebrasπ∗PnBΛ (n = 1, 2, . . . ). Instead, it is the seemingly innocuous choice of the sectionsn : KΛ → KΛ(Ωn+1Λ, n + 2), with which any twisted Eilenberg-Mac Lane object(of simplicial spaces or Π-algebras) is equipped:

8.5. Proposition. If X〈n〉 ∈ Rn(Λ) can be lifted to Rn(Λ), the possiblechoices for the (n+1)-st Postnikov stage X〈n + 1〉 are determined by the choicesof sections sn : KΛ → π∗X〈n〉, which correspond to elements in Hn+2(Λ; Ωn+1Λ).

8.6. The components. As noted above, the Dwyer-Kan concept of classifi-cation complexes has one advantage over the traditional moduli spaces, in that thetopology of each component encodes further information about the correspondinghomotopy type: namely, the topological group of self-equivalences.

MODULI SPACES OF HOMOTOPY THEORY 59

Such groups are generally hard to analyze, as one might expect from the topo-logical analogue (see, e.g., [44]). One advantage of the tower (7.10) is that thesuccessive fibers are generalized Eilenberg Mac-Lane spaces:

8.7. Proposition ([8, Prop. 9.6ff.]). For any Π-algebra Λ and n ≥ 0, the fiberof Rn+1(Λ) → Rn(Λ) is either empty, or has all components weakly equivalent to∏n+1

i=1 K(Hn+3−iΛ (Λ, Ωn+1Λ), i).

9. Comparing the rational and integral versions

The constructions of the rational moduli space as an infinite-dimensional alge-braic variety over Q, and of the integral moduli space as a classification complex,are not related in any evident way. However, given a Π-algebra Λ ∈ Π-Alg and itsrationalization ΛQ ∈ Π-AlgQ (defined ΛQ

1 := Λ1 and ΛQi := Λi ⊗ Q for i ≥ 2),

there is a connection between the set of components of MΛQ and that of MΛ,and also between the respective towers (7.10).

9.1. The preimage of rationalization. Clearly, Λ is realizable only if ΛQ is(which is automatic in the simply-connected case), but the converse need not hold(as the example of a non-realizable torsion Π-algebra in [3, Prop. 4.3.6] shows).Therefore, the number of components of MΛQ need have no relation to that ofMΛ, since there seems to be no known method for determining all the (integral)homotopy types X whose fibrewise rationalizations X ′

Q (§6.4I(b)) are homotopyequivalent. However, the obstruction theories of §2.10 and Propositions 8.1-8.5,respectively, do provide a framework for studying the preimage of the (fibrewise)rationalization:

Given a rationalized Π-algebra A, we have

(i) an algebraic question: which Π-algebras Λ have ΛQ ∼= A?(ii) a homotopy-theoretic question: what are the components of MΛ (though

different components can map to the same component of MA under thefibrewise rationalization functor)?

For the first question, restrict attention to the simply-connected case, so that Ais just a (connected) graded Lie algebra over Q. In this case we must first considerwhich connected graded Lie algebras over Z have L ⊗ Q ∼= A. Given such anL, we would like to classify all possible Π-algebra structures on L; one possibleapproach to this problem is given by the obstruction theory of [11], in terms ofcertain relative cohomology groups (where “relative” involves comparing the sameobject in different categories via a forgetful functor).

Theoretically, the obstruction theory of Section 8 provides an answer to thesecond question, although it does not tell us how to identify all integral homotopytypes having weakly equivalent rationalizations.

9.2. Comparing cohomology theories. In view of the obstruction theoriesof Sections 2 and 8, a first step towards understanding the relation between theintegral and rational classifications is to compare the cohomology theories thathouse the respective obstructions. There are two main cases to consider:

I. We can compare the cohomology of a Π-algebra Λ, with coefficients in someΛ-module M (say, M = ΩnΛ) with the cohomology of the rationalization ΛQ,

60 DAVID BLANC

with rationalized coefficients in MQ (which is a Λ-module, so in particular aΛQ-module).

In this case the exact sequence of Λ-modules 0 → Ki−→ M

q−→ MQ → C → 0

(for K := Ker (q) and C := Coker (q)) yields two long exact sequences incohomology, in the usual way. Moreover, the functors

Π-AlgT−→ Π-AlgQ S

−→ AbGp,

where T (−) := (−)Q and S(−) := HomΛ(−, MQ), satisfy the conditions of [12,

Thm. 4.4], since a rationalized free Π-algebra is free (and so H-acyclic) in Π-AlgQ.Thus we have a generalized Grothendieck spectral sequence with

(9.3) E2s,t = (LsSt)(L∗T )Λ ⇒ Hs+t(ΛQ; MQ)

converging to the cohomology in the category Π-AlgQ). Here Ls denotes the s-thleft derived functor, and (for simply-connected Λ) S : bg Lie → grAbGp is thefunctor induced by S, which exists because the homotopy groups of any simplicialgraded Lie algebra over Q actually take value in the category bg Lie of bigradedLie algebras.

For general Λ ∈ Π-Alg, by [12, Prop. 3.2.3] we have instead a functor S :

(Π-AlgQ)-Π-Alg → grAbGp whose domain is the analogue for Π-AlgQ of thecategory of Π-algebras for spaces. Its objects are bigraded groups, endowed withan action of all primary homotopy operations which exist for the homotopy groupsof a simplicial rational Π-algebra. As in the simply-connected case, these includethe bigraded Lie bracket mentioned above, and presumably others.

II. Alternatively, one could start with a rationalized Π-algebra A – for simplic-ity, a graded Lie algebra – and try to compare its cohomology (with coefficientsin an A-module M) taken in the category of Lie algebras with that obtained bythinking of it as an ordinary Π-algebra. This means taking the derived functors ofthe composite of

Π-AlgQ I−→ Π-Alg

S−→ AbGp,

where I is the inclusion, and S(−) := HomΠ-Alg/A(−, M) = HomΠ-Alg/I(A)(−, M).These do not satisfy the conditions of [12, Thm. 4.4], since a rationalized free

Π-algebra is not a free Π-algebra. However, we can take a simplicial resolutionV → A in the category sΠ-AlgQ of simplicial graded Lie algebras, and thenresolve each Vn functorially in sΠ-Alg to get a bisimplicial free Π-algebra W••,with Xd := diag W•• a free Π-algebra resolution of I(A). If Ab(−) denotes theabelianzation in the over category Π-Alg/A, then AbW•• is a bisimplicial abelianobject, or equivalently, a double complex in (Π-Alg/A)ab, and the bicosimplicialabelian group SW•• is HomΠ-Alg/A(AbW••, M). Since

HomΠ-Alg/A(X, M) = Hom(Π-Alg/A)ab(diag AbW••, M)

≃ TotHom(Π-Alg/A)ab(AbW••, M)

= TotHomΠ-Alg/A(W••, M)

by the generalized Eilenberg-Zilber Theorem, we see that both the spectral se-quences for the bicomplex SW•• converge to the cohomology of I(A). Since bydefinition

πtvSWn•

∼= Ht(IQn, M) = Ht(LnI(−), M)(A),

MODULI SPACES OF HOMOTOPY THEORY 61

we obtain a cohomological spectral sequence with

(9.4) Es,t2 = (πsHt(−, M))(L∗I)A ⇒ Hs+t(I(A), M).

This is less useful than (9.3), since we cannot identify the E2-term explicitlyas a derived functor.

9.5. Remark. As was pointed out in Section 4, another way to relate the inte-gral and rational moduli spaces is geometrically, using higher homotopy operations.This was the motivations behind [7] (in conjunction with [4]). The main difficultyin using homotopy operations in any systematic way for such a purpose is the lackof an appropriate taxonomy. The most promising way to overcome this wouldbe by establishing a clear correspondence between higher operations and suitablecohomology groups.

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Department of Mathematics, , University of Haifa, , 31905 Haifa, Israel

E-mail address: blanc@math.haifa.ac.il